Weak Solutions in Elasticity of Dipolar Porous Materialsdownloads.hindawi.com/journals/mpe/2008/158908.pdf · ijknm,A ijkmnr A mnrijk, a ij a ji,P ij P ji,g ij g ji. 2.6 The physical

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2008, Article ID 158908, 8 pagesdoi:10.1155/2008/158908

Research ArticleWeak Solutions in Elasticity of DipolarPorous Materials

Marin Marin

Department of Mathematics, University of Brasov, 500188 Brasov, Romania

Correspondence should be addressed to Marin Marin, [email protected]

Received 25 March 2008; Accepted 17 July 2008

Recommended by K. R. Rajagopal

The main aim of our study is to use some general results from the general theory of ellipticequations in order to obtain some qualitative results in a concrete and very applicative situation.In fact, we will prove the existence and uniqueness of the generalized solutions for the boundaryvalue problems in elasticity of initially stressed bodies with voids (porous materials).

Copyright q 2008 Marin Marin. This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.

1. Introduction

The theories of porous materials represent a material length scale and are quite sufficient fora large number of the solid mechanics applications.

In the following, we restrict our attention to the behavior of the porous solids inwhich the matrix material is elastic and the interstices are voids of material. The intendedapplications of this theory are to the geological materials, like rocks and soils and to themanufactured porous materials.

The plane of the paper is the following one. In the beginning, we write down thebasic equations and conditions of the mixed boundary value problemwithin context of lineartheory of initially stressed bodies with voids, as in the papers of [1, 2]. Then, we accommodatesome general results from the paper [3], and the book [4], in order to obtain the existence anduniqueness of a weak solution of the formulated problem. For convenience, the notationschosen are almost identical to those of [2, 5].

2. Basic equations

Let B be an open region of three-dimensional Euclidean space R3 occupied by our porousmaterial at time t = 0. We assume that the boundary of the domain B, denoted by ∂B,is a closed, bounded and pice-wise smooth surface which allows us the application of the

mailto:[email protected]

2 Mathematical Problems in Engineering

divergence theorem. A fixed system of rectangular Cartesian axes is used and we adopt theCartesian tensor notations. The points in B are denoted by (xi) or (x). The variable t is thetime and t ∈ [0, t0). We will employ the usual summation over repeated subscripts whilesubscripts preceded by a comma denote the partial differentiation with respect to the spatialargument. We also use a superposed dot to denote the partial differentiation with respect tot. The Latin indices are understood to range over the integers (1, 2, 3).

In the following, we designate by ni the components of the outward unit normal to thesurface ∂B. The closure of domain B, denoted by B, means B = B ∪ ∂B.

Also, the spatial argument and the time argument of a function will be omitted whenthere is no likelihood of confusion.

The behavior of initially stressed bodies with voids is characterized by the followingkinematic variables:

ui = ui(x, t), ϕjk = ϕjk(x, t), σ = σ(x, t), (x, t) ∈ B × [0, t0

). (2.1)

In our study, we analyze an anisotropic and homogeneous initially stressed elasticsolid with voids. We restrict our considerations to the Elastostatics, so that the basic equationsbecome as follows.

(i) The equations of equilibrium is as follows:

(τij + ηij

),j+ ρFi = 0,

μijk,i + ηjk + uj,iMik + ϕkiMji − ϕkr,iNijr + ρGjk = 0;(2.2)

(ii) the balance of the equilibrated forces is as follows:

hi,i + g + ρL = 0; (2.3)

(iii) the constitutive equations are as follows:

τij = uj,kPki + Cijmnεmn +Gmnijκmn + Fmnrijχmnr + aijσ + dijkσ,k,

ηij = −ϕjkMik + ϕjk,rNrik +Gijmnεmn + Bijmnκmn +Dijmnrχmnr + bijσ + eijkσ,k,

μijk = uj,rNirk + Fijkmnεmn +Dmnijkκmn +Aijkmnrχmnr + cijkσ + fijkmσ,m,

hi = dmniεmn + emniκmn + fmnriχmnr + diσ + gijσ,j ,

g = −amnεmn − bmnκmn − cmnrχmnr − ξσ + diσ,i;

(2.4)

(iv) the geometric equations are

εij =12(uj,i + ui,j

), κij = uj,i − ϕij ,

χijk = ϕjk,i, σ = ν − ν0.(2.5)

In the above equations we have used the following notations:

(i) ρ—the constant mass density;

(ii) ui—the components of the displacement field;

Marin Marin 3

(iii) ϕjk—the components of the dipolar displacement field;

(iv) ν—the volume distribution function which in the reference state is ν0;

(v) σ—a measure of volume change of the bulk material resulting from voidcompaction or distension;

(vi) τij , ηij , μij—the components of the stress tensors;

(vii) hi—the components of the equilibrated stress;

(viii) Fi—the components of body force per unit mass;

(ix) Gjk—the components of dipolar body force per unit mass;

(x) L—the extrinsic equilibrated body force;

(xi) g—the intrinsic equilibrated body force;

(xii) εij , κij , χijk—the kinematic characteristics of the strain tensors;

(xiii) Cijmn, Bijmn, . . . , Dijm, Eijm, . . . , aij , bij , cijk, di, ξ represent the characteristic functionsof the material (the constitutive coefficients) and they obey to the followingsymmetry relations

Cijmn = Cmnij = Cijnm, Bijmn = Bmnij ,

Gijmn = Gijnm, Fijkmn = Fijknm, Aijkmnr = Amnrijk,

aij = aji, Pij = Pji, gij = gji.

(2.6)

The physical significances of the functions L and hi are presented in the works [6, 7].The prescribed functions Pij , Mij and Nijk from (2.2) and (2.3) satisfy the following

equations:(Pij +Mij

),j= 0, Nijk,i + Pjk = 0. (2.7)

3. Existence and uniqueness theorems

In the main section of our paper, we will accommodate some theoretical results fromthe theory of elliptic equations in order to derive the existence and the uniqueness of ageneralized solution of the mixed boundary-value problem in the context of initially stressedbodies with voids.

Throughout this section, we assume that B is a Lipschitz region of the Euclidian three-dimensional space R3. We use the following notations:

W =[W1,2(B)

]13, W0 =

[W1,2

0 (B)]13, (3.1)

with the convention that A13 = A × A × · · · × A, the Cartesian product is considered to beof 13-times. Also, Wk,m is the familiar Sobolev space. With other words, W is defined as thespace of all u = (ui, ϕij , σ), where ui, ϕij , σ ∈W1,2(B)with the norm

|u|2W = |σ|2W1,2(B) +

3∑

i=1

∣∣ui∣∣2W1,2(B) +

3∑

j=1

( 3∑

i=1

∣∣ϕij∣∣2W1,2(B)

). (3.2)

For clarity and simplification in presentation, we consider the following regularityhypotheses on the considered functions:


(i) all the constitutive coefficients are functions of class C2 on B;

(ii) the body loads Fi, Gjk, andH are continuous functions on B.

The ordered array (ui, ϕjk, σ) is an admissible process on B = B ∪ ∂B provided ui, ϕjk,σ ∈ C1(B) ∩ C2(B). Also, the ordered array of functions (τij , ηij , μijk, hi) is an admissiblesystem of stress on B if τij , ηij , μijk, hi ∈ C1(B) ∩ C0(B) and τij,i, ηij,i, μijk,k, hk,k, h ∈ C0(B).

Let ∂B = Su ∪ St ∪ C be a disjunct decomposition of ∂B, where C is a set of surfacemeasure and Su and St are either empty or open in ∂B. Assume the following boundaryconditions:

ui = ui, ϕjk = ϕjk σ = σ on Su,

ti ≡(τij + ηij

)nj = ti, μjk ≡ μijkni = μjk, h ≡ hini = h on St,

(3.3)

where the functions ui, ϕjk, σ, ti, μjk, and h are prescribed, ui, ϕjk, σ ∈ W1,2(Su), and ti, μjk,h ∈ L2(St). Also, we define V as a subspace of the space W of all functions u = (ui, ϕij , σ)which satisfy the boundary conditions:

ui = 0, ϕij = 0, σ = 0 on Su. (3.4)

On the product space W ×W, we consider a bilinear form A(v,u), defined by

A(v,u) =∫

B

{Cijmnεmn(u)εij(v) +Gmnij

[εij(v)κmn(u) + εij(u)κmn(v)

]

+ Fmnrij[εij(v)χmnr(u) + εij(u)χmn(v)

]+ Bijmnκij(v)κmn(u)

+Dijmnr

[κij(v)χmnr(u) + κij(u)χmn(v)

]+Aijkmnrχijk(v)χmnr(u)

+ Pkiuj,kvj,i −Mik(uj,iψjk + vj,iϕjk) +Nrik(uj,kψjk,r + vj,kϕjk,r)

+ aij[εij(v)σ + εij(u)γ)

]+ bij

[κij(v)σ + κij(u)γ)

]

+ cijk[χijk(v)σ + χij(u)γ)

]+ dijk

[εij(v)σ,k + εij(u)γ,k)

]

+ eijk[κij(v)σ,k + κij(u)γ,k)

]+ fijkm

[χijk(v)σ,m + χijk(u)γ,m)

]

+ di[σγ,i + γσ,i

]+ gijσ,iγ,j + ξσγ

}dV,

(3.5)

where

u =(ui, ϕij , σ

), v =

(vi, ψij , γ

), εij(u) =

12(uj,i + ui,j

),

εij(v) =12(vj,i + vi,j

), κij(u) = uj,i − ϕij , κij(v) = vj,i − ψij ,

χijk(u) = ϕjk,i, χijk(v) = ψjk,i.

(3.6)

We assume that the constitutive coefficients are bounded measurable functions in B whichsatisfy the symmetries (2.6). Then, by using relations (3.5) and (2.6) it is easy to deduce that

A(v,u) = A(u,v). (3.7)

Marin Marin 5

Also, by using symmetries (2.6) into (3.5), it results in

A(u,u) =∫

B

[Cijmnεmn(u)εij(v) + 2Gmnijεij(u)κmn(u)

+ Bijmnκij(u)κmn(u) + 2Fmnrijεij(u)χmnr(u) + 2Dijmnrκij(u)χmnr(u)

+Aijkmnrχijk(u)χmnr(u) + Pkiuj,kuj,i − 2Mikuj,iϕjk

+ 2Nrikuj,iϕjk,r + 2aijεij(u)σ + 2bijκij(u)σ + 2cijkχijk(u)σ

+ 2dijkεij(u)σ,k + 2eijkκij(u)σ,k + 2fijkmχijk(u)σ,m

+ 2diσγ,i + gijσ,iσ,j + ξσ2]dV,

(3.8)

and thus

A(u,u) = 2∫

B

UdV, (3.9)

whereU = ρe is the internal energy density associated to u.We suppose that U is a positive definite quadratic form, that is, there exists a positive

constant c such that

Cijmnxijxmn + 2Gijmnxijymn + 2Fijmnrxijzmnr

+ Bijmnyijymn + 2Dijmnryijzmnr +Aijkmnrzijkzmn

+ Pkixjixjk − 2Mikxjiyjk + 2Nrikxjizjkr + 2aijxijw

+ 2bijyijw + 2cijkzijw + 2dijkxijωk + 2eijkyijωk

+ 2fijkmzijkωm + 2diwωi + gijkωiωj + ξw2

≥ c(xijxij + yijxij + zijkzijk +ωiωi +w2),

(3.10)

for all xij , yij , zijk, ωi, and w.Now, we introduce the functionals f(v) and g(v) by

f(v) =∫

B

ρ(Fivi +Gjkψjk + Lγ

)dV, v ∈ W,

g(v) =∫

St

(tivi + μjkψjk + hγ

)dA, v ∈ W,

(3.11)

where v = (vi, ψjk, γ) ∈ W and ρ, Fi, Gjk, L ∈ L2(B).Let v = (ui, ϕjk, γ) ∈ W be such that ui, ϕjk, γ on Su may by obtained by means of

embedding the spaceW1,2 into the space L2(Su).The element v = (ui, ϕjk, σ) ∈ W is called weak (or generalized) solution of the boundary

value problem, if

u − u ∈ V,

A(u,u) = f(u) + g(v)(3.12)


hold for each v ∈ V. In the above relations, we used the spaces L2(B) and L2(Su) whichrepresent, as it is well known, the space of real functions which are square-integrable on B,respectively, on Su ⊂ ∂B.

It follows from (3.10) and (3.8) that

A(v,v) ≥ 2c∫

B

[εij(v)εij(v) + κij(v)κij(v) + χijk(v)χijk(v) + γiγi + γ2

]dV, (3.13)

for any v = (vi, ψjk, γ) ∈ W.Let us consider the operatorsNkv, k = 1, 2, . . . , 49, mapping the spaceW into the space

L2(B), defined by

Niv = ε1i(v), N3+iv = ε2i(v), N6+iv = ε2i(v),

N9+iv = κ1i(v), N12+iv = κ1i(v), N15+iv = κ1i(v),

N18+iv = χ11i(v), N21+iv = χ12i(v), N24+iv = χ13i(v),



N45+iv = σ,i(v), N49v = σ(v) (i = 1, 2, 3).

(3.14)

It is easy to see that, in fact, the operatorsNkv, k = 1, 2, . . . , 49, defined above, have thefollowing general form:

Nkv =m∑

r=1

∑

|α|≤krnkrαD

αvr, p = |α|, (3.15)

where nkrα are bounded and measurable functions on B. Also, we have used the notation Dα

for the multi-indices derivative, that is,

Dα =∂|α|

∂xα11 ∂xα22 ∂x

α33

. (3.16)

By definition, the operatorsNkv, (k = 1, 2, . . . , 49) form a coercive system of operatorson Wif for each v ∈ W the following inequality takes place:

49∑

k=1

∣∣Nkv∣∣2L2(B)

+13∑

r=1

∣∣vr∣∣2L2(B)

≥ c1|v|2W, c1 > 0. (3.17)

In this inequality, the constant c1 does not depend on v and the norms |·|L2and |·|W

represent the usual norms in the spaces L2(B) and W, respectively.In the following theorem, we indicate a necessary and sufficient condition for a system

of operators to be a coercive system.

Theorem 3.1. Let npsα be constant for |α| = ks. Then the system of operatorsNpv is coercive onW ifand only if the rank of the matrix

(Npsξ

)=( ∑

|α|=ksnpsαξα

)(3.18)

is equal tom for each ξ ∈ C3, ξ /= 0, where C3 is the notation for the complex three-dimensional space,and

ξα = ξα11 ξα22 ξ

α33 . (3.19)

Marin Marin 7

The demonstration of this result can be find in [4].In the following, we assume that for each v ∈ W, we have

A(v,v) ≥ c249∑

k=1

∣∣Nkv∣∣2L2, c2 > 0, (3.20)

where the constant c2 does not depend on v.We denote by P the following set:

P =

{

v ∈ V :49∑

k=1

∣∣Nkv∣∣2L2

= 0

}

, (3.21)

and by V/P the factor-space of classes v, where

v ={v + p, v ∈ V, p ∈ P}

, (3.22)

having the norm

∣∣v∣∣V/P = inf

p∈P|v + p|W. (3.23)

In the following theorem, it is indicated a necessary and sufficient condition for theexistence of a weak solution of the boundary-value problem.

Theorem 3.2. Let A(v,u) = [v, u] define a bilinear form for each v, u ∈ W/P, where u ∈ u andv ∈ v. If it is supposed that the inequalities (3.17) and (3.20) hold, then a necessary and sufficientcondition for the existence of a weak solution of the boundary value problem is

p ∈ P =⇒ f(p) + g(p) = 0. (3.24)

Moreover, the weak solution, u ∈ W, satisfies the following inequality:

|u|W/P ≤ c3[∣∣u∣∣W +

(m∑

i=1

∣∣fi∣∣L2(B)

)1/2

+

(m∑

i=1

∣∣gi∣∣L2(S)

)1/2]

, (3.25)

where c3 is a real positive constant.Further, one has

A(v, v

) ≥ c4∣∣v∣∣W/P, c4 > 0, (3.26)

for each v ∈ W/P.

For the prove of this result, see [3].In the following, we intend to apply the above two results in order to obtain the

existence of a weak solution for the boundary value problem formulated in the context oftheory of initially stressed elastic solids with voids.

Theorem 3.3. LetP = {0}. Then there exists one and only one weak solution u ∈ W of our boundary-value problem.


Proof. Clearly, from (3.13) and (3.14)we immediately obtain (3.20). The matrix (3.20) has therank 13 for each ξ ∈ C3, ξ /= 0. Thus by Theorem 3.1 we conclude that the system of Nk

operators, defined in (3.14), is coercive on the space W.According to definition (3.21) of P, we have that εij(v) = 0, κij(v) = 0, χijk(v) = 0,

γi(v) = 0, ψ = 0 for each v ∈ P, v = (vi, ψjk, ψ).So, we deduce that P reduces to

P ={v = (vi, ψjk, γ) ∈ V : vi = ai + εijkbjxk, ψjk = εjksbs, γ = c

}, (3.27)

where ai and bi and c are arbitrary constants and εijk is the alternating symbol.Wewill consider two distinct cases. First, we suppose that the set Su is nonempty. Then

the setP reduces toP = {0}, and therefore, condition (3.24) is satisfied. By using Theorem 3.2,we immediately obtain the desired result.

In the second case, we assume that Su is an empty set. Then we have the followingresult.

Theorem 3.4. The necessary and sufficient conditions for the existence of a weak solution u ∈ W ofthe boundary-value problem for elastic dipolar bodies with stretch, are given by

∫

B

ρFidV +∫

∂B

ti dA = 0,

∫

B

ρεijk(xjFk +Gjk

)dV +

∫

∂B

εijk(xj tk + μjk

)dA = 0,

(3.28)

where εijk is the alternating symbol.

Proof. In this case, the boundary value problem P is given by (3.27), where ai, bi, and c arearbitrary constants such that we can apply, once again, Theorem 3.2 to obtain the above result.

4. Conclusion

For the considered initial-boundary value problem the basic results still valid. Now, fordifferent particular cases, the solution can be found because it exists and is unique.

References

[1] D. Iesan, “Thermoelastic stresses in initially stressed bodies with microstructure,” Journal of ThermalStresses, vol. 4, no. 3-4, pp. 387–398, 1981.

[2] M. Marin, “Sur l’existence et l’unicite dans la thermoelasticite des milieux micropolaires,” ComptesRendus de l’Academie des Sciences. Serie II, vol. 321, no. 12, pp. 475–480, 1995.

[3] I. Hlavacek and J. Necas, “On inequalities of Korn’s type. I: boundary-value problems for ellipticsystem of partial differential equations,” Archive for Rational Mechanics and Analysis, vol. 36, no. 4, pp.305–311, 1970.

[4] I. Necas, Les Methodes Directes en Theorie des Equations Elliptiques, Academia, Prague, Czech Republic,1967.

[5] M. Marin, “On the nonlinear theory of micropolar bodies with voids,” Journal of Applied Mathematics,vol. 2007, Article ID 15745, 11 pages, 2007.

[6] M. A. Goodman and S. C. Cowin, “A continuum theory for granular materials,” Archive for RationalMechanics and Analysis, vol. 44, no. 4, pp. 249–266, 1972.

[7] J. W. Nunziato and S. C. Cowin, “Linear elastic materials with void,” Journal of Elasticity, vol. 13, pp.125–147, 1983.

Submit your manuscripts athttp://www.hindawi.com

Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttp://www.hindawi.com

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation http://www.hindawi.com Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttp://www.hindawi.com

Volume 2014 Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

Stochastic AnalysisInternational Journal of

Documents

Weak Solutions in Elasticity of Dipolar Porous Materialsdownloads.hindawi.com/journals/mpe/2008/158908.pdf · ijknm,A ijkmnr A mnrijk, a ij a ji,P ij P ji,g ij g ji. 2.6 The physical